Quadratic Unconstrained Binary Optimization (QUBO) formulations have been proposed as a pathway for applying quantum optimization algorithms to fault tree analysis. These formulations encode fault tree logic through penalty terms, converting constrained minimal cut set identification into unconstrained optimization. Prior work has demonstrated QUBO-MCS equivalence for small, carefully structured instances. This paper investigates whether penalty-based encodings of the type commonly used in the literature reliably preserve fault tree semantics at realistic problem scales when solved exactly. A cohort of 80 extended fault tree instances with OR, AND, and NOT gates was evaluated using two exact solvers: a constrained baseline that enforces fault tree semantics directly, and a branch-and-bound solver operating on soft QUBO formulations using Rosenberg product reductions. Even with the TOP event variable fixed, penalty coefficients conservatively scaled to exceed the maximum objective contribution, and exact (not heuristic) optimization, the soft QUBO solutions matched the constrained baseline in only 6 of 80 instances (7.5%). More critically, 22 of 80 instances (27.5%) produced solutions that violate fault tree gate semantics entirely. These results demonstrate that Rosenberg-based soft QUBO formulations with dominance-safe penalties do not reliably preserve fault tree semantics at scale under exact optimization, indicating a structural limitation of this class of unconstrained penalty encodings rather than a tuning deficiency. Complete experimental artifacts with SHA-256 verification are provided under the DOI above.
Quadratic Unconstrained Binary Optimization (QUBO) formulations have been proposed as a pathway for applying quantum optimization algorithms to fault tree analysis. These formulations encode fault tree logic through penalty terms, converting constrained minimal cut set identification into unconstrained optimization. Prior work has demonstrated QUBO-MCS equivalence for small, carefully structured instances. This paper investigates whether penalty-based encodings of the type commonly used in the literature reliably preserve fault tree semantics at realistic problem scales when solved exactly. A cohort of 80 extended fault tree instances with OR, AND, and NOT gates was evaluated using two exact solvers: a constrained baseline that enforces fault tree semantics directly, and a branch-and-bound solver operating on soft QUBO formulations using Rosenberg product reductions. Even with the TOP event variable fixed, penalty coefficients conservatively scaled to exceed the maximum objective contribution, and exact (not heuristic) optimization, the soft QUBO solutions matched the constrained baseline in only 6 of 80 instances (7.5%). More critically, 22 of 80 instances (27.5%) produced solutions that violate fault tree gate semantics entirely. These results demonstrate that Rosenberg-based soft QUBO formulations with dominance-safe penalties do not reliably preserve fault tree semantics at scale under exact optimization, indicating a structural limitation of this class of unconstrained penalty encodings rather than a tuning deficiency. Complete experimental artifacts with SHA-256 verification are provided under the DOI above.
Quadratic Unconstrained Binary Optimization formulations for fault tree analysis encode gate logic through penalty terms, but prior work has demonstrated that these encodings fail to preserve fault tree semantics at scale: across 80 extended fault tree instances, only 7.5% of Rosenberg-based soft QUBO solutions match the constrained exact baseline, while 27.5% produce outright semantic violations. This paper develops two complementary methods to restore semantic correctness. A Hierarchical Semantic Validator and repair framework achieves 100% semantic validity across all 80 cohort instances through structure-aware bit-flip correction, improving the exact match rate from 7.5% to 28.75%. A constraint-level QUBO encoding eliminates penalty-based failures entirely by restricting optimization to the feasible subspace, reducing qubit requirements by a factor of 1.85x relative to Rosenberg encodings. Validation through Qiskit mixer circuits implementing a feasibility-preserving Quantum Alternating Operator Ansatz confirms zero probability leakage onto infeasible states for representative instances. These methods establish that semantic-preserving quantum optimization for fault tree minimal cut set identification is achievable with current algorithmic tools. Complete experimental artifacts with SHA-256 verification are provided under the DOI above.
This paper presents a formal validation framework establishing equivalence between Quadratic Unconstrained Binary Optimization (QUBO) formulations and classical minimal cut set (MCS) identification for fault tree analysis in probabilistic risk assessment. The framework develops a 21-gate hierarchical verification system that proves QUBO ground states correspond exactly to minimal cut sets computed by the SCRAM fault tree solver. The validation methodology employs exhaustive enumeration for tractable models and conservative spot checks for larger instances, achieving PASS status across all verification gates. The theoretical contribution formalizes MCS identification as a constrained Boolean satisfiability problem amenable to quantum optimization. Fault trees with OR-of-ANDs structure are encoded into QUBO form with penalty terms enforcing logic constraints. The encoding preserves semantic equivalence: valid QUBO ground states map bijectively to classical minimal cut sets. Empirical validation demonstrates the framework on a canonical test case (TOP = A ∨ B ∨ (C ∧ D)) using QAOA simulation with plateau-based convergence methodology. The quantum circuit correctly identifies valid MCS configurations with p_opt = 0.01058 (95% CI: [0.00972, 0.01151]). Hardware feasibility is demonstrated through execution on IBM's ibm_torino 133-qubit Heron processor, where all circuits passed sanity and interpretation checks despite noise distributing probability across 110-120 unique states. The framework establishes foundational infrastructure for quantum PRA research by providing: (1) formal QUBO-MCS bijection verification against ground truth, (2) a reproducible classical baseline using the SCRAM solver, (3) plateau-based convergence criteria for stochastic quantum optimization, and (4) a hierarchical verification gate system ensuring scientific defensibility. No claim of quantum advantage is made; results are evaluated solely for semantic correctness, reproducibility, and NISQ feasibility.
This paper evaluates whether a native constrained formulation, rather than a penalty-style QUBO conversion, helps D-Wave execution pathways recover fault-tree truth-objective solutions, using the locked 80-instance benchmark cohort from a prior platform-feasibility study (Paper 17), stratified into four problem-size groups of twenty instances each (16, 25, 33, and 42 Paper 17 QUBO variables). Two execution arms operate on the same cohort and the same frozen truth layer. Arm A is a direct-QPU pilot on a constraint-level QUBO (CL-QUBO)-derived penalty-style binary quadratic model (BQM). Arm B is a native constrained quadratic model (CQM) submitted to the D-Wave hybrid CQM solver, in which the fault-tree gate constraints are preserved as explicit constraints rather than compiled into penalties. The truth layer uses a minimum-cardinality criterion: an instance is truth-positive when a returned feasible sample matches the canonical truth objective. On the canonical eight-instance pilot, the direct-QPU Arm A recovered truth on 4 of 8 instances, with zero-truth behavior in the larger pilot strata, whereas the native hybrid CQM Arm B recovered truth on 8 of 8. Arm B recovery extended to a balanced twelve-instance set (12 of 12) and to the full locked cohort, recovering a truth-positive feasible solution on all 80 instances, with zero zero-truth instances and zero objective-below-truth anomalies, including 20 of 20 in every size stratum. This paper does not claim quantum advantage, computational speedup, or isolated QPU contribution within the hybrid solver; because the two arms differ in both formulation and pathway, it does not claim to isolate formulation as the sole causal factor. A retrospective timing audit of the preserved execution artifacts, performed without any resubmission, recovered complete solver timing metadata for all 88 jobs and found that two 16-variable instances were solved entirely by classical presolve with zero QPU access time, which bounds the interpretation of the full-cohort result without changing any recovery count. The results support semantic preservation as a central design requirement: the native constrained CQM pathway recovered the truth objective across the full cohort, while the penalty-style direct-QPU pilot showed size-limited recovery under the tested settings.
Fault-tree analysis is a core component of probabilistic risk assessment (PRA), and quadratic unconstrained binary optimization (QUBO) provides a direct interface for evaluating fault-tree optimization problems on quantum annealers. A prior size-stratified study of an 80-instance fault-tree cohort observed sharp degradation in direct quantum-processing-unit (QPU) minimum-cardinality truth recovery as QUBO size increased, but the direct-QPU arms used a single anneal duration and did not isolate embedding, chain-strength, logical-variable relabelling, or chain-break explanations. This follow-on study prospectively pre-specified and SHA-256-locked a controlled robustness experiment on the same frozen penalty-QUBO cohort using D-Wave Advantage2 hardware. Eighty instances spanning 16, 25, 33, and 42 QUBO variables were evaluated across five forward-anneal durations and three deterministic fixed embeddings, with additional chain-strength controls on chained embeddings and four deterministic logical-variable relabelling diagnostics at 42 variables. The campaign completed 1,408 of 1,408 pre-specified conditions without execution or artifact failures and preserved 1,408,000 physical and 1,408,000 logical reads. Parameter selection affected recovery at smaller sizes: all 16- and 25-variable instances were truth-positive under at least one core condition. In contrast, only 18 of 20 33-variable instances and 3 of 20 42-variable instances were ever recovered across the fifteen anneal-time × embedding conditions, so 17 of 20 42-variable instances were never truth-positive. No tested anneal-time or chain-strength condition met the pre-specified material-rescue criterion, and four 42-variable relabelling diagnostics recovered zero instances. Only one of 1,408,000 logical reads exhibited a nonzero chain-break fraction, and that read was infeasible and not truth-positive. The results show size-dependent tuning leverage rather than parameter invariance: ordinary direct-QPU controls improve recovery within the smaller-size regime but do not materially rescue the large-size recovery frontier on this cohort and hardware epoch. Because the design fixes read count rather than QPU access time, anneal-duration comparisons are interpreted under equal read budgets rather than equal time. The study does not claim quantum advantage, universal formulation failure, full minimal-cut-set enumeration, or any time-to-solution or speedup result.
This paper evaluates whether feasible-subspace-preserving variational circuit families can compete with the frozen QAOA+ baseline from Peters (2026) for Constraint-Level QUBO (CL-QUBO) fault tree minimal cut set identification. Six alternative constructions—an independently parameterized XX+YY generator family, a cost-phase-augmented variant, a transpiled-matched variant, a structure-ordered variant, an n = 6 restricted screen, and a warm-start variant—were tested against the QAOA+ comparator on a frozen pilot cohort of 14 reactor-scale fault tree subtrees (10 n = 6, 4 n = 8a instances) under noiseless statevector simulation with deterministic multi-restart optimization. The directly materialized family constructions passed feasible-subspace closure audits, and the screen-only variants inherited closure-qualified source families. These audits confirmed that the excitation-number-conserving XX+YY construction remained inside the intended fixed-Hamming-weight sector for the tested CL-QUBO instances. The failure mechanism is identified through per-instance analysis: the alternative families concentrate essentially all statevector probability (≈99.99%) on a single computational basis state that does not correspond to a minimal cut set, while QAOA+ distributes probability across the feasible subspace with sufficient structure to place MCS among the highest-ranked outcomes. The independently parameterized generator family lacks the problem-specific cost–mixer alternation that gives QAOA+ its concentration advantage, and adding a single cost phase layer does not rescue this deficit. The result is a disciplined negative finding: within the tested family definitions and frozen screening protocol, no approved alternative produced evidence strong enough to justify hardware execution. The contribution is a careful narrowing of the viable variational design space for CL-QUBO fault tree optimization, establishing that feasible-subspace preservation alone is insufficient for competitive MCS enrichment. Complete experimental artifacts with SHA-256 verification are provided.
Heterogeneous HPC workflow scheduling under multiple hard constraints poses a challenging combinatorial optimization problem. Classical exact solvers provide optimal solutions but often face scalability limitations, motivating interest in quantum-inspired Quadratic Unconstrained Binary Optimization (QUBO) formulations as alternative optimization paradigms. This work presents a systematic and reproducible empirical evaluation of QUBO-based scheduling methods against established classical baselines, including MILP, CP-SAT, GA, and HEFT. We evaluate three QUBO variants, single-run simulated annealing, multi-attempt annealing, and a layered QAOA-inspired schedule, together with hybrid enhancement strategies on both ground-truth validation workflows (3–4 tasks) and synthetic scaling instances (5–20 tasks). All solvers are assessed through a unified evaluation pipeline that explicitly tracks feasibility, makespan, and resource utilization under progressive constraint activation and controlled penalty sweeps. Results show that all approaches recover the expected optimal makespan on validation instances, confirming formulation correctness. However, feasibility degradation emerges for specific QUBO variants as constraint interactions intensify, particularly when communication costs are introduced. Penalty sensitivity analysis reveals a sharp feasibility threshold for QUBO-SA, where insufficient penalties consistently fail and moderate-to-strong penalties restore feasibility. Scaling experiments delineate the regimes in which classical solvers and heuristics remain robust across all tested sizes, while QUBO-SA loses feasibility beyond 15 tasks and the QAOA-inspired variant beyond 10 tasks. Overall, the study provides a clear empirical characterization of the reliability boundaries of quantum-inspired QUBO formulations for heterogeneous HPC scheduling and identifies the regimes where classical approaches remain preferable under current formulations and solver capabilities.